Resummation of Double-Differential Cross Sections and Fully-Unintegrated Parton Distribution Functions

LHC measurements involve cuts on several observables, but resummed calculations are mostly restricted to single variables. We show how the resummation of a class of double-differential measurements can be achieved through an extension of Soft-Collinear Effective Theory (SCET). A prototypical application is $pp \to Z + 0$ jets, where the jet veto is imposed through the beam thrust event shape ${\mathcal T}$, and the transverse momentum $p_T$ of the $Z$ boson is measured. A standard SCET analysis suffices for $p_T \sim m_Z^{1/2} {\mathcal T}^{1/2}$ and $p_T \sim {\mathcal T}$, but additional collinear-soft modes are needed in the intermediate regime. We show how to match the factorization theorems that describe these three different regions of phase space, and discuss the corresponding relations between fully-unintegrated parton distribution functions, soft functions and the newly defined collinear-soft functions. The missing ingredients needed at NNLL/NLO accuracy are calculated, providing a check of our formalism. We also revisit the calculation of the measurement of two angularities on a single jet in JHEP 1409 (2014) 046, finding a correction to their conjecture for the NLL cross section at ${\mathcal O}(\alpha_s^2)$.